Quick Truth Table in Logic Problem

Question

Suppose We Have:

How can quickly detect how many "1" are in the truth table of above formula? (without drawing Truth Table).

i think by using some inference. any idea?

we know there are 11 "1"s in TT.

Answer 1

Presumably one is only interested in $1$s in the output.

There are $8$ where $p_4$ is true. Or else we want the antecedent of $p_4$ false, forcing $p_3$ false, and $p_1\longrightarrow p_2$ true. The count of cases where $p_1\longrightarrow p_2$ is true, and the rest false is easy.

One uses a similar analysis in producing a "quick" disjunctive normal form. I don't know whether this is what problem-setter has in mind.

Another way: It can be useful to count instead the ways to make a sentence false. In our case we want $p_4$ false and its antecedent true. The antecedent is true if $p_3$ is true ($4$ cases, since then the values of $p_1$ and $p_2$ are arbitrary) or if $p_3$ is false and $p_1\longrightarrow p_2$ is false ($1$ case). That gives a total of $5$. Subtract from $16$.